Question:
The curve with equation y = 3x^2 - 2x + 5 is the graph of a function f(x). Consider the portion of the curve between x = 0 and x = 4.
a) Find the length of the curve over this interval using the arc length formula. Show all steps.
b) Without using the arc length formula, find an approximation for the length of the curve over the interval [0, 4] by dividing the interval into four equal subintervals and approximating the curve with straight line segments. Show all steps.
Answer:
a) To find the length of the curve using the arc length formula, we need to integrate the square root of the sum of the squares of the derivatives dy/dx with respect to x.
The derivative of y = 3x^2 - 2x + 5 is:
dy/dx = 6x - 2
Now, we need to find √(1 + (dy/dx)^2). Let's substitute dy/dx into this formula:
√(1 + (dy/dx)^2) = √(1 + (6x - 2)^2)
To find the arc length, we need to integrate the expression above from x = 0 to x = 4:
L = ∫[(0 to 4)] √(1 + (6x - 2)^2) dx
To solve this integral, let's make a substitution: let u = 6x - 2. Then, du = 6 dx and dx = du/6.
Now we can rewrite the integral: L = ∫[(0 to 4)] √(1 + u^2) (du/6)
L = (1/6)∫[(0 to 4)] √(1 + u^2) du
To evaluate this integral, we can use a trigonometric substitution. Let's let u = tan(t), then du = sec^2(t) dt. The limits of integration also change.
When u = 0, t = 0, and when u = 4, t = tan^(-1)(4).
Now our integral becomes: L = (1/6)∫[(0 to tan^(-1)(4))] √(1 + tan^2(t)) sec^2(t) dt
Simplifying inside the square root: L = (1/6)∫[(0 to tan^(-1)(4))] √(sec^2(t)) sec^2(t) dt
L = (1/6)∫[(0 to tan^(-1)(4))] sec^3(t) dt
This integral can be evaluated using the power rule for integrals: ∫ sec^n(t) dt = (1/n) sec^(n-1)(t) tan(t) + (n-1)/n ∫ sec^(n-2)(t) dt
Applying the power rule, we get: L = (1/6)[(1/3) sec^2(t) tan(t) + (2/3) ∫ sec(t) dt] from 0 to tan^(-1)(4)
To evaluate this expression, we need the integral of sec(t), which is ln|sec(t) + tan(t)|.
L = (1/6)[(1/3) sec^2(t) tan(t) + (2/3) ln|sec(t) + tan(t)|] evaluated from 0 to tan^(-1)(4)
L = (1/6)[(1/3) sec^2(tan^(-1)(4)) tan(tan^(-1)(4)) + (2/3) ln|sec(tan^(-1)(4)) + tan(tan^(-1)(4))| - (1/3) sec^2(0) tan(0) - (2/3) ln|sec(0) + tan(0)|]
Notice that sec^2(0) = 1 and tan(0) = 0, simplifying the expression:
L = (1/6)[(1/3) sec^2(tan^(-1)(4)) tan(tan^(-1)(4)) + (2/3) ln|sec(tan^(-1)(4)) + tan(tan^(-1)(4))|]
Using the trigonometric identity sec(tan^(-1)(x)) = √(x^2 + 1), we can simplify the expression even further:
L = (1/6)[(1/3)(4^2 + 1) tan(tan^(-1)(4)) + (2/3) ln|√(4^2 + 1) + 4|]
L = (1/6)[(17/3) tan(tan^(-1)(4)) + (2/3) ln(√(17) + 4)]
Finally, we can evaluate tan(tan^(-1)(4)): tan(tan^(-1)(4)) = 4
Substituting this back into our expression for L: L = (1/6)[(17/3)(4) + (2/3) ln(√(17) + 4)]
Simplifying: L = (4/3)(17 + ln(√17 + 4))
Therefore, the length of the curve over the interval [0, 4] is (4/3)(17 + ln(√17 + 4)).
b) To find an approximation for the length of the curve over the interval [0, 4] without using the arc length formula, we will divide the interval into four equal subintervals and approximate the curve with straight line segments.
Divide the interval [0, 4] into four subintervals: [0, 1], [1, 2], [2, 3], and [3, 4].
Using the given equation y = 3x^2 - 2x + 5, we can find the equation of a straight line segment connecting the endpoints of each subinterval.
For the subinterval [0, 1]: P1(0, 5) and P2(1, 6), slope = (6 - 5) / (1 - 0) = 1, equation of the line segment: y = x + 5.
For the subinterval [1, 2]: P1(1, 6) and P2(2, 11), slope = (11 - 6) / (2 - 1) = 5, equation of the line segment: y = 5x + 1.
For the subinterval [2, 3]: P1(2, 11) and P2(3, 20), slope = (20 - 11) / (3 - 2) = 9, equation of the line segment: y = 9x - 7.
For the subinterval [3, 4]: P1(3, 20) and P2(4, 33), slope = (33 - 20) / (4 - 3) = 13, equation of the line segment: y = 13x + 7.
We can find the lengths of these line segments using the distance formula: For the first line segment: L1 = √[(1 - 0)^2 + (6 - 5)^2] = √2 For the second line segment: L2 = √[(2 - 1)^2 + (11 - 6)^2] = √26 For the third line segment: L3 = √[(3 - 2)^2 + (20 - 11)^2] = √100 = 10 For the fourth line segment: L4 = √[(4 - 3)^2 + (33 - 20)^2] = √314
The total approximation for the length of the curve is the sum of the lengths of these line segments: Approximate length = L1 + L2 + L3 + L4 = √2 + √26 + 10 + √314
Therefore, the approximate length of the curve over the interval [0, 4] is √2 + √26 + 10 + √314.